Common omissions and misconceptions of wave propagation in turbulence: discussion

doi:10.1364/JOSAA.29.000711

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Common omissions and misconceptions of wave propagation in turbulence: discussion

Mikhail Charnotskii »View Author Affiliations

JOSA A, Vol. 29, Issue 5, pp. 711-721 (2012)
http://dx.doi.org/10.1364/JOSAA.29.000711

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Abstract

This review paper addresses typical mistakes and omissions that involve theoretical research and modeling of optical propagation through atmospheric turbulence. We discuss the disregard of some general properties of narrow-angle propagation in refractive random media, the careless use of simplified models of turbulence, and omissions in the calculations of the second moment of the propagating wave. We also review some misconceptions regarding short-exposure imaging, propagation of polarized waves, and calculations of the scintillation index of the beam waves.

1. INTRODUCTION

This paper is motivated by a pattern of misconceptions, omissions, and mistakes that developed in the literature concerning the optical wave propagation in turbulent atmosphere. We see this critical review as a resource for the authors and reviewers, and as an opportunity to raise awareness about some available fundamental theoretical results, which are not well represented in the latest books.

Section 2 discusses the general properties of narrow-angle propagation in refractive random media: energy conservation and reciprocity. These very general properties are valid for individual realizations of the propagating optical field, and thus are not related to any specific turbulent model. We will show that they have very important consequences for turbulence propagation, but are mostly overlooked or misinterpreted in the current literature.

Section 3 addresses some issues related to specific models of turbulence that are often used in the literature. We show how the use of the oversimplified Gaussian spectral model of turbulence delivers totally erroneous results for the beam scintillation index (SI). We discuss the meaning and potential dangers of the quadratic structure function (QSF) for modeling of turbulent perturbations. We disclose some common errors in representation of the calculation results for non-Kolmogorov turbulence.

Section 4 discusses some lapses and errors that are commonly made when calculating the parameters related to the second statistical moment of the propagating wave. We address the unnecessary use of the Huygens–Fresnel and Rytov approximations for calculations of average beam intensity. We reflect on the meaning of the mean-square beam size as applied to turbulence-distorted beams. We point to a simple, but completely overlooked, theoretical result: average beam irradiance is a convolution of undistorted beam irradiance with the turbulent point spread function (PSF). We briefly review propagation of polarized optical waves through turbulence and expose the inadequacy of the popular model of electromagnetic beam propagation in turbulence and explain how the calculation results have been misinterpreted.

In Section 5 we discuss a series of misconceptions that are common in SI calculations. We show using several examples that a single parameter, e.g., so-called Rytov variance, is not sufficient to characterize the magnitude of turbulence impact on propagating optical waves. These examples logically lead to the introduction of the plane of dimensionless parameters, where the boundary between “weak” and “strong” turbulence can be sensibly drawn. We clarify the infamous misunderstanding of the Rytov’s approximation (vanishing scintillation at the beam focus), and we show the correct weak and strong scintillation solutions for the SI at the beam focus.

In Section 6 we discuss the flaws of the Fried model of short-term PSF, and focus on the more accurate PSF model.

It is not our intension to criticize any specific papers. We are more concerned about the general patterns that have arisen in the recent literature. Therefore, we do not cite the particular papers where the flaws we discuss are present. We hope our paper could be used as a reference in future work.

2. GENERAL PROPERTIES OF NARROW-ANGLE OPTICAL PROPAGATION

Throughout the paper, we use a Cartesian coordinate system with

z

as the principal propagation direction. For laser beam propagation the source is located in the plane

z = 0

, and light propagates to the observation plane

z = L > 0

. For imaging problems, the object is located in the plane

z = L > 0

, and light propagates toward the receiving aperture located in the plane

z = 0

. Optical wave propagation through turbulence with refractive index

1 + \tilde{n} (r, z)

is described by the parabolic equation [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]

\pm 2 i k \frac{\partial}{\partial z} u (r, z) + Δ u (r, z) + {2 k}^{2} \tilde{n} (r, z) u (r, z) = 0.

(1)

Here, the plus sign corresponds to a wave propagating in the positive

z

direction, and the minus to a wave propagating in the opposite direction. Given the initial condition

u (r, z_{1})

, the solution of Eq. (1) can be presented as

u (r_{2}, z_{2}) = \iint d^{2} r_{1} u (r_{1}, z_{1}) G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2}),

(2)

where

G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2})

is the Green’s function for propagation from plane

z = z_{1}

to plane

z = z_{2}

. Obviously, the Green’s function is just a field of the spherical wave from a point source located at

(r_{1}, z_{1})

, which is observed at the point

(r_{2}, z_{2})

. For the free-space (no inhomogeneity) case

G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2}) = \frac{k}{2 π i | z_{1} - z_{2} |} \exp (\frac{i k {(r_{1} - r_{2})}^{2}}{2 | z_{1} - z_{2} |}) .

(3)

In general case, the solution for the Green’s function exists only in the form of the Feynman path integral [2

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics , Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.

], but some features of the Green’s function are well known, and can be used even when a tractable solution is not available.

A. Reciprocity

One of these features is reciprocity. Reciprocity is an essential property of the Maxwell equations, and for the simpler case of the parabolic Eq. (1) it was discussed in [3

V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]

]. In terms of the Green’s function (3), reciprocity is presented as

G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2}) = G (r_{2}, z_{2} \Rightarrow r_{1}, z_{1}) .

(4)

It was shown in [4

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]

] that the presence of an optical system in the form of a series of apertures and phase screens preserves the reciprocity. One frequently overlooked benefit of reciprocity is that it allows using the solution found for one propagation problem to solve a completely different problem without any additional efforts. We present two simple examples here.

Consider power flux

P

through the aperture at

z = L

with irradiance transmittance function

O (R)

from a point source at the origin. Using the Green’s function formalism we write

P = \iint d^{2} R O (R) I (R) = \iint d^{2} R O (R) {| G (0, 0 \Rightarrow R, L) |}^{2} .

(5)

On the other hand, irradiance at the point (0,0) from an incoherent source with luminosity

O (R)

positioned at

z = L

I (0, 0) = \iint d^{2} R O (R) {| G (R, L \Rightarrow 0, 0) |}^{2} .

(6)

Applying reciprocity to Eq. (5) we find that the power flux

P

and irradiance

I (0, 0)

fluctuate completely in synchronism, and any statistics calculated for the power flux can be used directly for the irradiance.

The second example is the axial irradiance of the focused beam. The axial field in the focal plane of the beam with focal distance

L

and real amplitude

A (r)

at the initial plane

z = 0

can be written as

u (0, L) = \iint d^{2} r A (r) \exp (- \frac{i k}{L} r^{2}) G (r, 0 \Rightarrow 0, L) .

(7)

The initial field at the plane

z = 0

can be presented as a field of the point source positioned at

(- l, 0)

that free-space propagated to

z = 0

, and passed through the aperture

A (r)

and a thin lens with focal distance

f = {(l^{- 1} + L^{- 1})}^{- 1}

. This presents the field at the focus as

u (0, L) = C \iint d^{2} r G (0, - l \Rightarrow r, 0) \times A (r) \exp [- i k (\frac{1}{l} + \frac{1}{L}) r^{2}] G (r, 0 \Rightarrow 0, L) .

(8)

Note that the first Green’s function in Eq. (8) is given by Eq. (3). Using reciprocity we find that

u (0, L)

is also the field in the image of the point source located at

(0, L)

and imaged through the aperture A(r). Similar to the first example, optical fields in the image and beam focus fluctuate identically. The results for the beam irradiance statistics can be directly applied to the image statistics. This was used in [5

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

], where the axial SI of the focused beam was interpreted in terms of the image scintillation.

Besides extending the theoretical results, reciprocity can provide a way of monitoring the optical field or irradiance in a remote inaccessible plane. Indeed, reciprocity between the focused beam and the image of a target allows monitoring of the power that can be delivered to the target at any given moment simply by watching the irradiance of the target image. More detailed discussion of this idea can be found in [4

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]

B. Energy Conservation

The basic feature of the Green’s function of the parabolic equation (1) is superposition. Specifically, using Eq. (2) twice for

z_{1} < z_{2} < z_{3}

or for

z_{1} > z_{2} > z_{3}

, we have

G (r_{1}, z_{1} \Rightarrow r_{3}, z_{3}) = \iint d^{2} r_{2} G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2}) \times G (r_{2}, z_{2} \Rightarrow r_{3}, z_{3}) .

(9)

Equation (9) allows composing the Green’s function for a longer propagation path from two Green’s functions for the path segments.

The next property of the Green’s function is closely related to energy conservation [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]. Consider a parabolic equation for the instantaneous coherence function

γ (R, ρ, z) = u (R + ρ / 2, z) u^{*} (R - ρ / 2, z)

that follows from Eq. (1) after multiplying Eq. (1) for

u (r_{1}, z)

u^{*} (r_{2}, z)

, subtracting from it Eq. (1) for

u^{*} (r_{2}, z)

multiplied by

u (r_{1}, z)

and changing vector arguments as indicated in the definition of the coherence function above

\pm i k \frac{\partial}{\partial z} γ (R, ρ, z) + \nabla_{R} \cdot \nabla_{ρ} γ (R, ρ, z) + k^{2} [\tilde{n} (R + \frac{ρ}{2}, z) - \tilde{n} (R - \frac{ρ}{2}, z)] γ (R, ρ, z) = 0 .

(10)

Total power in the propagating beam wave at the plane

z

can be calculated as

P (z) = \iint d^{2} R γ (R, ρ, z) |_{\vec{ρ} = 0} .

(11)

Applying the integration over variable

R

to Eq. (10), we use a divergence theorem to eliminate the second term. Setting

ρ = 0

eliminates the third term, and Eq. (10) reduces to

\frac{\partial}{\partial z} P (z) = 0 .

(12)

In other words for any

z

, total power remains constant. The Green’s function representation of this result is

\iint d^{2} R u (R, z_{1}) u^{*} (R, z_{1}) = \iint d^{2} r_{1} \iint_{1} d^{2} r_{2} u (r_{1}, z_{1}) u^{*} (r_{2}, z_{1}) \iint d^{2} R G (r_{1}, 0 \Rightarrow R, z_{2}) G^{*} (r_{2}, 0 \Rightarrow R, z_{2}) .

(13)

This should be valid for any field distribution

u (R, z_{1})

; hence

\iint d^{2} R G (r_{1}, 0 \Rightarrow R, z_{2}) G^{*} (r_{2}, 0 \Rightarrow R, z_{2}) = δ (r_{1} - r_{2}) .

(14)

Equation (14) represents the orthogonal property of the Green’s functions [6

V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).

]. Orthogonality is the theoretical “backbone” of the conjugated wave propagation technique. On the other hand in [7

V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).

] it was shown that orthogonality explains the absence of scintillation in the image of the uniformly illuminated objects observed through refractive turbulence. Simulation and modeling of the turbulent image distortions use computer-generated random PSFs. We are unaware of any models that account for energy conservation. As a result simulated images show erroneous scintillation in the areas with low contrast. Path-integral derivation of Eq. (14) can be found in the appendix of [8

M. I. Charnotskii, “Turbulence effects on the imaging of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996). [CrossRef]

Energy conservation also imposes some restriction on the irradiance covariance

b_{I} (R, ρ)

. For a bounded beam wave, the following equation constrains possible covariance functions [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

\iint d^{2} {R d}^{2} ρ b_{I} (R, ρ) = 0 .

(15)

For the unbounded, statistically uniform waves, of which plane and spherical waves are examples, energy conservation implies that [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]

\iint d^{2} ρ b_{I} (ρ) = 0 .

(16)

Equation (16) affects the rate at which scintillation is averaged by a finite collecting aperture. It was discussed in [9

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991). [CrossRef]

,10

M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).

] that for large collecting apertures “superaveraging” takes place, and “power-in-a-bucket” fluctuation decreases faster with the aperture size than predicted by simple statistical averaging. Equations (15) and (16) also imply that there are positions and separations where scintillations are anticorrelated. This can be beneficial for the space diversity mitigation of fading in the free-space optical communication systems, if receiving apertures are positioned at the negative correlation separation.

Combining Eqs. (9) and (14) for

z_{1} < z_{2} < z_{3}

or for

z_{1} > z_{2} > z_{3}

, we arrive at another interesting property of the Green’s function:

G (r_{1}, z_{1} \Rightarrow r_{2}, z_{2}) = \iint d^{2} r_{2} G (r_{1}, z_{1} \Rightarrow r_{3}, z_{3}) \times G^{*} (r_{3}, z_{3} \Rightarrow r_{2}, z_{2}) .

(17)

While Eq. (9) shows how to add a new propagation segment to the propagating wave, Eq. (17) allows reversal of the propagation process.

Equations (9) and (14)–(17) as well as reciprocity equation (4) have to be satisfied by any empirical propagation model, e.g., the Huygens–Fresnel approximation, or scintillation theory [11

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).

], but this is rarely a concern for scientists who develop and use these models. Another sensitive area is the propagation of the partially coherent beams. As was discussed in [4

V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]

,12

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010). [CrossRef]

], partial coherence is closely related to virtual incoherent sources, and a correct model for the irradiance averaging is crucial for accurate models of the scintillation in the partially coherent beams. However, it seems that these issues never receive attention in the relevant literature.

3. TURBULENCE MODELS

All calculations of the statistical parameters of the optical waves propagating through turbulence depend on the spectrum of the fluctuations of the refractive index

Φ_{n} (κ, z)

. The simplest and the most widely used analytical form of the power spectrum of the turbulent fluctuations of the refractive index is the Kolmogorov spectrum [13

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Φ_{n} (κ, z) = 0.033 C_{n}^{2} (z) κ^{- 11 / 3} .

(18)

This spectrum is self-similar (fractal) and has no intrinsic scale and no variance.

A. Gaussian Spectrum

Some of the earlier works involving propagation in turbulence used the Gaussian spectral model. Unlike the Kolmogorov spectrum (18), this model is characterized by two parameters variance

σ_{n}

and scale

l_{n}

of the refractive index fluctuations

Φ_{n} (κ) = \frac{σ_{n}^{2} l_{n}^{3}}{8 π \sqrt{π}} \exp (- \frac{κ^{2} l_{n}^{2}}{4}) .

(19)

The Gaussian model is very convenient for analytical calculations. However, use of the Gaussian spectrum as a proxy for the turbulence spectrum generates results that have nothing to do with the turbulence propagation. We illustrate this in a simple example.

In the simplest first-order perturbation approximation, the statistics of the propagating wave are linearly related to the spectrum. For example, the axial SI of the collimated Gaussian beam with initial field

u (r, z = 0) = \exp (- \frac{r^{2}}{2 a^{2}})

(20)

is given by the following well-known equation that can be found, for example, in [5

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

σ_{I}^{2} = {8 π k}^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ, z) \exp [- \frac{κ^{2} a^{2}}{(1 + N^{2})} {(1 - \frac{z}{L})}^{2}] \sin^{2} [\frac{κ^{2} a^{2} N}{2 (1 + N^{2})} {(1 - \frac{z}{L})}^{2} + κ^{2} \frac{z}{2 k} (1 - \frac{z}{L})] .

(21)

There are three multiplicative terms in the spectral integral in Eq. (21). The first term is the turbulent spectrum itself; the second, exponential, term has the typical scale related to the effective beam width. The last, sinusoidal, term is responsible for the diffraction of scattered light, and has the typical scale related to the beam width and the Fresnel radius. The relations between the scales of the exponential and sinusoidal terms are independent of the spectrum

Φ_{n} (κ, z)

and are governed by a single dimensionless parameter—the Fresnel number for the initial beam size

N = \frac{k a^{2}}{L} .

(22)

The Kolmogorov spectrum is “passive” in Eq. (21), which means that it does not define the region of the main contribution to the spectral integral, but delegates this task to other terms under the integral. As a result the asymptotic structure of the SI is very simple: For

N < 1

, when the beam wave is similar to the spherical wave,

σ_{I}^{2} \approx 8 π k^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ, z) \sin^{2} [κ^{2} \frac{z}{2 k} (1 - \frac{z}{L})] = 2.25 k^{7 / 6} L^{11 / 6} \int_{0}^{1} d t C_{n}^{2} (L t) t^{5 / 6} {(1 - t)}^{5 / 6} = O (q^{- 5 / 6}),

(23)

and for

N > 1

, when the beam wave is similar to a plane wave,

σ_{I}^{2} \approx 8 π k^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ, z) \sin^{2} [κ^{2} \frac{L}{2 k} (1 - \frac{z}{L})] = 2.25 k^{7 / 6} L^{11 / 6} \int_{0}^{1} d t C_{n}^{2} (L t) {(1 - t)}^{5 / 6} = O (q^{- 5 / 6}),

(24)

where

q = \frac{k r_{C}^{2}}{L}, r_{C} = {[1.46 k^{2} L \int_{0}^{1} C_{n}^{2} (L t) {(1 - t)}^{5 / 3} d t]}^{- 3 / 5}

(25)

is the Fresnel number for the spherical wave coherence radius

r_{C}

. Direct calculations show that for narrow beams

N \propto 1

there are no dramatic changes of the SI, and

σ_{I}^{2} = O (q^{- 5 / 6})

uniformly in

N

The situation is very different for the Gaussian spectrum (19). The intrinsic turbulence scale

l_{n}

of the first term in Eq. (21) now competes with the variable scales of the second and third terms. This creates the possibility of more complex asymptotic structure of the SI as a function of dimensionless arguments represented by the Fresnel numbers

N

and

M \equiv k l_{n}^{2} / 4 L

. The superficial benefit of the Gaussian spectrum is that the spectral domain integration in Eq. (23) can be performed analytically with the following result:

σ_{I}^{2} \approx 8 \sqrt{π} \frac{k^{2} l_{n}^{3} L}{{2 a}^{2}} \int_{0}^{1} \frac{d t σ_{n}^{2} (L t) D (t)}{C (t) [C^{2} (t) + D^{2} (t)]}, C (t) \equiv \frac{M}{N} + \frac{{(1 - t)}^{2}}{{1 + N}^{2}}, D (t) \equiv \frac{t (1 - t)}{N} + \frac{N {(1 - t)}^{2}}{{1 + N}^{2}} .

(26)

However, asymptotic analysis of

σ_{I}^{2} (N, M)

reveals the following six asymptotic domains:

σ_{I}^{2} = \frac{\sqrt{π}}{2} {(k l_{n})}^{3} \int_{0}^{1} d t σ_{n}^{2} (L t) t^{2} {(1 - t)}^{2} \frac{1}{M^{3}}, M > 1 > N, σ_{I}^{2} = \frac{\sqrt{π}}{2} {(k l_{n})}^{3} \int_{0}^{1} d t σ_{n}^{2} (L t) {(1 - t)}^{2} \frac{1}{M^{3}}, M > 1 > 1 / N, σ_{I}^{2} = \frac{\sqrt{π}}{2} {(k l_{n})}^{3} \int_{0}^{1} d t σ_{n}^{2} (L t) \frac{1}{M}, (1 > M > 1 / N) \cup (1 > M > N), σ_{I}^{2} = \frac{π \sqrt{π}}{4} {(k l_{n})}^{3} σ_{n}^{2} (L) \sqrt{\frac{N}{M}}, 1 < N < 1 / M, σ_{I}^{2} = \frac{π \sqrt{π}}{4} {(k l_{n})}^{3} σ_{n}^{2} (L) \sqrt{\frac{1}{M N}}, M < N < 1.

(27)

While similar to the Kolmogorov spectrum case, SI in most cases is proportional to the weighted path-integrated variance of the refractive index fluctuations; the wave and path length dependencies do not match the Kolmogorov case for any combination of parameters. The Gaussian spectrum results describe completely different propagation physics and cannot be used as a proxy for the turbulent spectrum.

B. Quadratic Structure Function

Turbulent structure function

D (\vec{ρ})

emerges in a variety of propagation models, including Rytov’s smooth perturbation approximation, phase screen, and the Huygens–Fresnel method. All these models in fact are certain approximations to the fundamental Markov approximation [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]. The Markov approximation makes it possible to derive a parabolic equation for the propagation of the mean coherence function by averaging Eq. (10)

\pm 2 i k \frac{\partial}{\partial z} Γ + (Δ_{1} - Δ_{2}) \nabla_{R} \cdot \nabla_{ρ} Γ + i \frac{π k^{3}}{2} H (ρ, z) Γ (R, ρ, z) = 0,

(28)

where

Γ (R, ρ, z) = 〈 γ (R, ρ, z) 〉

is the mean coherence function, and

H (ρ, z) = 4 \iint d^{2} κ Φ_{n} (κ, z) [1 - \cos (κ \cdot ρ)] .

(29)

Parabolic equation (28) has a relatively simple analytic solution [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

Γ (R, ρ, L) = \frac{k^{2}}{4 π^{2} L^{2}} \iint d^{2} R_{0} d^{2} ρ_{0} Γ_{0} (R_{0}, ρ) \exp [\frac{i k}{L} (R - R_{0}) (ρ - ρ_{0}) - \frac{1}{2} D (ρ, ρ_{0})],

(30)

where the two-argument structure function

D (ρ, ρ_{0}) = \frac{π k^{2}}{2} \int_{0}^{L} d z H [ρ_{0} (1 - \frac{z}{L}) + ρ \frac{z}{L}, z]

(31)

contains all the information about the turbulence on the propagation path. For the mean irradiance calculations, only

D (ρ_{0}) \equiv D (ρ = 0, ρ_{0})

is necessary. For the Kolmogorov spectrum (18),

D (ρ_{0}) = 2 {(\frac{ρ_{0}}{r_{C}})}^{5 / 3},

(32)

where coherence radius

r_{C}

was defined in Eq. (25). It is very common in the literature to replace the

5 / 3

structure function with the QSF

D (ρ_{0}) \approx 2 {(\frac{ρ_{0}}{r_{C}})}^{2},

(33)

whereas, in order to relate the calculations to turbulence, the definition of the coherence radius

r_{C}

remains the same as in Eq. (25). In many cases the QSF simplifies the analytic calculations to a great extent. The issues arising from QSF have been discussed in the literature [14

S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980). [CrossRef]

], but nonetheless we discuss some remarkable features of the QSF here.

Consider homogeneous random field

S (r)

with structure function (33). The local slope of this field is

\nabla S (r) = (S_{x}, S_{y})

. The covariance matrix of the vector random field of slopes is

〈 \nabla S (r + ρ) \nabla S (r) 〉 = (\begin{matrix} D_{x x} (ρ) & D_{x y} (ρ) \\ D_{x y} (ρ) & D_{y y} (ρ) \end{matrix}) .

(34)

For QSF (33), the covariance matrix of slopes is

2 r_{C}^{- 2} I_{2}

, where

I_{2}

is the

2 \times 2

identity matrix. This implies that for each realization of

S (r)

, slopes are constant for any point in the plane. In other words, each realization is a linear function of coordinates.

The same conclusion can be derived if we consider the error of the statistical interpolation of

S (r)

. For simplicity we limit ourselves to the one-dimensional case

r = x \hat{x}

. Assume that instantaneous values of

S (x)

are perfectly measured at two points:

x_{1}

and

x_{2}

. We seek the estimate

\tilde{S} (x)

S (x)

in the form of the linear combination of the measured values

\tilde{S} (x) = a S (x_{1}) + b S (x_{2}),

(35)

and find the interpolation coefficients

a

and

b

that minimize the interpolation error

Δ_{S}^{2} = 〈 {[S (x) - \tilde{S} (x)]}^{2} 〉

. Imposing a condition of zero partial derivatives of

Δ_{S}^{2}

with respect to

a

and

b

determines the values of these coefficients, and the optimal interpolation equation is specified from Eq. (35) as

\tilde{S} (x) = \frac{1}{2} [(1 - \frac{D (x - x_{1}) - D (x - x_{2})}{D (x_{1} - x_{2})}) S (x_{1}) + (1 - \frac{D (x - x_{2}) - D (x - x_{1})}{D (x_{1} - x_{2})}) S (x_{2})] .

(36)

The interpolation error can be calculated as

Δ_{S}^{2} = \frac{1}{4 D (x_{1} - x_{2})} [2 D (x_{1} - x_{2}) (D (x_{1}) - D (x_{2})) - {(D (x_{1}) - D (x_{2}))}^{2} - D^{2} (x_{1} - x_{2})] .

(37)

It is easy to check that for QSF, interpolation coefficients are linear functions of

x

, and interpolation error

Δ_{S}^{2} \equiv 0

. Thus, linear interpolation (or extrapolation) of

S (x)

from any two measurements absolutely accurately predicts its value at any point. This brings up the earlier conclusion: each realization of the random field with QSF is a linear function of coordinates.

For a general power-law one-dimensional structure function

D (x) \approx {(\frac{| x |}{r_{C}})}^{α},

(38)

the extrapolation error can be presented as follows:

Δ_{S}^{2} = \frac{D (x_{1})}{4 {| 1 - t |}^{α}} [2 {| 1 - t |}^{α} + 2 {| t |}^{α} {| 1 - t |}^{α} - {({1 - | t |}^{α})}^{2} - {| 1 - t |}^{2 α}], t = \frac{x_{1}}{x_{2}} .

(39)

Figure 1 shows the dependence of the bracket in Eq. (39) on

t

and

α

. The extrapolation error is nonnegative for

0 < α < 2

, identically zero for

α = 2

, and nonpositive for

α > 2

. The last case clearly indicates that structure functions with

α > 2

contradict the spatial homogeneity requirement, and are unphysical. This example also places QSF in a very peculiar position at the very “edge” of the set of realistic power-type structure functions.

Fig. 1. Dependence of the normalized extrapolation error on the point position for different exponents

α

of the power-law structure function (33).

Another interesting feature of the QSF is revealed from its relation to the turbulence spectrum. For QSF, Eqs. (31) and (29) necessitate that

\frac{1}{2} D (ρ) = \frac{ρ^{2}}{r_{C}^{2}} = π k^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ, z) [1 - \cos (κ \cdot ρ (1 - \frac{z}{L}))] .

(40)

The only way to satisfy Eq. (40) is to require that

Φ_{n} (κ, z) = A (z) Δ δ (κ), \int_{0}^{L} d z A (z) {(1 - \frac{z}{L})}^{2} = \frac{1}{π k^{2} r_{C}^{2}},

(41)

where

Δ δ (κ)

is the Laplacian of the (two-dimensional) 2D Dirac delta function. This is definitely not a suitable model for turbulence, but it describes the medium consisting of infinitely wide random wedges distributed along the propagation path. This type of propagation model was recently discussed in more detail in [15

G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006). [CrossRef]

,16

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203 (2009).

QSF does not work well for the scintillation calculations. Consider, for example, the parabolic equation for the fourth-order coherence function of a plane or spherical wave [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]

2 i k \frac{\partial}{\partial z} Γ_{4} + 2 \nabla_{1} \cdot \nabla_{2} Γ_{4} + i \frac{π k^{3}}{2} [2 H (r_{1}, z) + 2 H (r_{2}, z) - H (r_{1} + r_{2}, z) - H (r_{1} - r_{2}, z)] Γ_{4} (r_{1}, r_{2}, z) = 0,

(42)

where for the statistically uniform wave fluctuation it is sufficient to consider a specific configuration of the arguments of the fourth-order coherence function

Γ_{4} (r_{1}, r_{2}, z) = 〈 u (\frac{r_{1} + r_{2}}{2}, z) u (- \frac{r_{1} + r_{2}}{2}, z) u^{*} (\frac{r_{1} - r_{2}}{2}, z) u^{*} (- \frac{r_{1} - r_{2}}{2}, z) 〉 .

(43)

Clearly, for the QSF model, the turbulence-related term in parabolic equation (42) vanishes, and there is no scintillation for the plane and spherical waves. There are scintillations for the QSF model in the more general case of a bounded beam wave [15

G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006). [CrossRef]

,17

M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).

], but they are completely determined by beam wander.

C. Non-Kolmogorov Spectra

There is renewed interest in the literature in propagation in non-Kolmogorov turbulence. In most cases the non-Kolmogorov spectrum is prescribed in the form similar to Eq. (18) but with different exponent

Φ_{n} (κ, z) = \frac{Γ (α + 1)}{4 π^{2}} \sin (\frac{π}{2} (α - 1)) C_{n}^{2} (z) κ^{- 2 - α}, 1 < α < 2 .

(44)

Structure constant

C_{n}^{2}

has dimension

{(length)}^{1 - α}

. On several occasions the calculation results for non-Kolmogorov turbulence have been presented in the form of dependence, let us say SI, on the numerical values of

C_{n}^{2}

for different exponents

α

. This is a very misleading method of presentation since the relative position of the curves depends on the length units used for

C_{n}^{2}

. We illustrate this in Fig. 2, where identical SI calculation results for the spherical wave with wavelength 0.63 μm, and path length 1 km, are presented as a function of

C_{n}^{2}

using meters, centimeters, and millimeters as length units. The meter unit chart [Fig. 2(a)] suggests that the smaller spectral exponent causes stronger scintillation in weak scintillation conditions, but the millimeter unit chart in Fig. 2(c) gives the opposite impression. For the centimeter unit chart [Fig. 2(b)], SI is about the same in weak scintillation conditions for all

α

Fig. 2. Dependence of the spherical wave SI for the wavelength 0.63 μm and path length 1 km on the structure constant. Heavy solid line,

α = 5 / 3

; short dashed line,

α = 1.4

; and long dashed line,

α = 1.9

The correct way to present this type of comparison is to use a dimensional argument of the same dimension, e.g., coherence radius

r_{C}

[Eq. (25)], or some dimensionless argument, e.g., Fresnel number

q

or “Rytov variance”

σ_{I}^{2} = O (q^{- 5 / 6})

. Figure 3 presents the same data as does Fig. 2, but plotted as a function of coherence radius in Fig. 3(a), parameter

q

in Fig. 3(b), and

σ_{I}^{2}

in Fig. 3(b). While the unit choice inconsistency has been eliminated in these charts, there still is no single answer to the question about the SI dependence on the spectral exponent. The answer depends on the choice of parameters that are kept constant, while varying the spectral exponent

α

Fig. 3. Dependence of the spherical wave SI for the wavelength 0.63 μm, and path length 1 km on the coherence radius parameter

q

and “Rytov variance”. Heavy solid line,

α = 5 / 3

; short dashed line,

α = 1.4

; and long dashed line,

α = 1.9

4. SECOND MOMENT OF OPTICAL WAVES IN TURBULENCE

A. Huygens–Fresnel Method

There are dozens of recent papers in which average irradiance of the propagating beam is calculated following the same routine for various beam types: fully and partially coherent, higher-order Gaussian, flat-top, beam arrays, vortex beams, etc. In all cases, given the coherence function of the beam in the initial plane

z = 0

, say

Γ_{0} (R_{0}, ρ_{0}) = \bar{u_{0} (R_{0} + ρ_{0} / 2) u_{0}^{*} (R_{0} - ρ_{0} / 2)}

, where the over bar is used to indicate the averaging over the internal source fluctuation, the instantaneous irradiance at the observation plane

z = L

is written as a coherent superposition of the spherical waves

I (R) = \frac{k^{2}}{4 π^{2} L^{2}} \iint d^{2} R_{0} d^{2} ρ_{0} Γ_{0} (R_{0}, ρ_{0}) \exp [\frac{i k}{2 L} (R_{0} - R) \cdot ρ_{0} + i ψ (R_{0} + \frac{ρ_{0}}{2}, R) - i ψ (R_{0} - \frac{ρ_{0}}{2}, R)],

(45)

where complex phase

ψ (r, R)

is the turbulence-caused perturbation of the spherical wave propagating from point

(r, 0)

to point

(R, L)

. The averaging in Eq. (45) is performed using either phase-only perturbation and the simple geometrical optics approximation, or the first and second orders of Rytov’s smooth perturbation theory. In any case, unless some additional approximations are made, the average irradiance is presented as

〈 I (R) 〉 = \frac{k^{2}}{4 π^{2} L^{2}} \iint d^{2} R^{'} d^{2} ρ^{'} Γ_{0} (R^{'}, ρ^{'}) \exp [\frac{i k}{2 L} (R^{'} - R) \cdot ρ^{'} - \frac{1}{2} D (ρ^{'})],

(46)

where wave structure function

D (ρ^{'})

is given by Eqs. (29) and (31) with

ρ_{0} = 0

It is frequently overlooked that Eq. (46) is valid under much more general assumptions that are routinely made to derive it. In fact it is a specific case of the solution (30) to the parabolic equation (28) of the Markov approximation theory for propagation of the average coherence function [1

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

]. Equation (46) is easily recognized as a specific case of Eq. (30). There is no need to burden the calculation by all the assumptions associated with Rytov’s theory, when a more general and rigorous foundation is available.

B. Convolution Property

A very useful, but completely ignored, property of the mean irradiance given by Eq. (46) is revealed when this equation is presented in the spectral form. Taking the 2D Fourier transform of Eq. (46), we have

〈 \hat{I} (κ) 〉 = \frac{1}{4 π^{2}} \iint d^{2} R 〈 I (R) 〉 \exp (- i κ \cdot R) = {\hat{I}}_{FS} (κ) \exp [- \frac{1}{2} D (κ \frac{L}{k})],

(47)

where

{\hat{I}}_{FS} (κ) = \frac{1}{{4 π}^{2}} \iint d^{2} R_{0} Γ_{0} (R_{0}, \frac{κ L}{k}) \exp (i κ \cdot R_{0})

(48)

is the spatial spectrum of the free-space (no turbulence) irradiance distribution in the plane

z = L

. Equation (47) presents the average irradiance as a spatial filtering of the free-space irradiance by a filtering function universal for all propagating beams with the same wavelength. The spatial domain equivalent of Eq. (47) is the convolution integral

〈 I (R) 〉 = \iint d^{2} R_{1} I_{FS} (R_{1}) P (R - R_{1}),

(49)

where the turbulent PSF

P (R)

P (R) = \frac{1}{{4 π}^{2}} \iint d^{2} κ \exp (- \frac{1}{2} D (κ \frac{L}{k}) - i R \cdot κ) .

(50)

There are two remarkable features of Eq. (49):

• Average irradiance depends only on the undisturbed irradiance distribution in the observation plane. The overall beam shape for $0 < z < L$ does not matter. This is despite the fact that turbulence is distributed along the path and interacts with the beam for all $0 < z < L$ . In particular the full or partial coherence of the beam source makes no difference as long as the free-space irradiance at $z = L$ is the same.
• The turbulent PSF (50) is exactly the same as for the long-exposure imaging of incoherent objects through turbulence on the same propagation path.

While for the incoherent imaging, the existence of the PSF is a trivial consequence of the source spatial incoherence, the emergence of the PSF for the propagation of the coherent beam wave is totally unexpected. The convolution form of the average irradiance is still valid even when there is a thin lens placed somewhere on the propagation path, but only when the lens is much larger than the beam size at the lens plane, and diffraction on the lens aperture can be neglected. On a practical note, Eqs. (47) and (50) make calculation of the average irradiance a trivial task even for the most complicated model beams. Direct and inverse 2D Fourier transforms are all that is needed.

C. RMS Beam Size

From the earliest years of turbulence propagation studies, turbulent beam spread was one of the most common effects discussed in the literature. Beam spread in most cases is related to the average irradiance distribution discussed in the previous section. Modern propagation theory provides very accurate and not very complex tools for mean irradiance calculations. However, in many cases, researchers are looking not for a detailed beam profile, but for a simple metric characterizing the beam spread. Using an analogy with the simple Gaussian beam, a multitude of papers define the beam size for an arbitrary irradiance distribution

I (R)

as the RMS beam widths:

a_{RMS}^{2} = \frac{\iint R^{2} d^{2} R I (R)}{\iint d^{2} R I (R)} .

(51)

Indeed, for a Gaussian beam (20) with

e^{- 1}

half-width

a

, Eq. (51) gives

a_{RMS} = a

The situation is very different for the mean irradiance distribution (46). For Kolmogorov spectrum (18), the structure function

D (ρ)

is given by Eq. (32). Inspection of the integral in the nominator of Eq. (51) with

I (R)

given by Eq. (46) reveals that

\iint R^{2} d^{2} R 〈 I (R) 〉 = \frac{- L^{2}}{{4 π}^{2} k^{2}} Δ_{ρ} {[{\hat{I}}_{FS} (\frac{k}{L} ρ) \exp (- \frac{1}{2} D (ρ))]}_{ρ = 0} .

(52)

For

D (\vec{ρ}) \propto ρ^{5 / 3}

, the right-hand part of Eq. (52) diverges when

ρ \to 0

. This formal observation has a very simple explanation. For

R > L / k r_{C}

, the turbulent PSF (50) with structure function (32) has the following asymptote:

P (R) \approx {(2 π)}^{- 3 / 2} Γ (\frac{19}{6}) {(\frac{L}{k r_{c}})}^{5 / 3} R^{- 11 / 3} .

(53)

Equation (53) reveals that the turbulent PSF (50), and therefore any mean irradiance distribution, has relatively slow falloff at the periphery of the beam. Figure 4 shows numerically integrated PSF (50) and

r^{- 11 / 3}

dependence. A slow, power-law PSF tail is related to the abundance of the small-scale turbulent eddies in the Kolmogorov spectrum (18) that causes relatively wide-angle diffraction. One can rightfully argue that introduction of the inner scale

l_{0}

into the turbulence spectrum will resolve the divergence issue. This is correct, and it can be shown that for

R > L / {k l}_{0}

, irradiance falloff becomes Gaussian and the integral in the nominator converges. As a result, however, the

a_{RMS}

is proportional to

l_{0}^{- 1 / 6}

even for the very wide beams, where the inner scale effects should be negligible.

Fig. 4. Turbulent PSF for Kolmogorov spectrum. Dashed line,

r^{- 11 / 3}

dependence.

All these issues are related to the definition of the RMS beam size, which emphasizes the peripheral parts of the beam. In our opinion the Strehl number definition of the beam size

a_{ST}^{2} = a_{FS}^{2} \frac{I_{FS} (0)}{〈 I (0) 〉}

(54)

is a more appropriate metric for the central, energy-carrying part of the unimodal beams. Definition of the beam size for beams with more complex irradiance distributions is a separate issue, but RMS size is not a solution in these cases either.

Plugging QSF in the PSF equation (50) produces a Gaussian PSF. By increasing the coherence radius defined by Eq. (21) by 26%, it is possible to reproduce the central part of the PSF very accurately. The corresponding PSF shapes are shown in Fig. 5. The

R^{- 11 / 3}

“tails” are absent for the QSF, of course.

Fig. 5. Turbulent PSF for Kolmogorov spectrum (18) and QSF (33).

D. Polarized Beam Waves and Beam Arrays

In recent years, there was an exceptionally large number of papers presenting calculations of what authors referred to as electromagnetic beam waves propagation through turbulence. The investigations never extended beyond the calculations of the second moment of the propagating waves. The main conclusion was that the turbulence seriously affects the polarization of electromagnetic beam waves. This result is in stark contrast with the classic result from the dawn of turbulence propagation theory [13

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

] stating that for optical wavelengths, depolarization by atmospheric turbulence is negligibly small. We discussed this issue in greater detail in [18

M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010). [CrossRef]

], and will give just a short overview here.

Writing the vector electric field as

u = u_{1} \hat{x} + u_{2} \hat{y}

, and assuming propagation in the positive direction of the

z

-axis, we present the simplest equation that accounts for depolarization of the short-wave polarized electrical field as follows:

2 i k \frac{\partial u_{1} (r, z)}{\partial z} + Δ_{r} u_{1} + 2 k^{2} \tilde{n} (r, z) u_{1} = - 2 \frac{\partial}{\partial x} (u_{2} \frac{\partial \tilde{n}}{\partial y}), 2 i k \frac{\partial u_{2} (r, z)}{\partial z} + Δ_{r} u_{2} + 2 k^{2} \tilde{n} (r, z) u_{2} = - 2 \frac{\partial}{\partial y} (u_{1} \frac{\partial \tilde{n}}{\partial x}), r = x \hat{x} + y \hat{y} .

(55)

The right-hand part of these equations describes depolarization, i.e., excitation of the cross-polarized component by scattering on the turbulent inhomogeneities, and is proportional to the gradients of inhomogeneity. Simple scale analysis reveals that for wavelengths shorter than the turbulence inner scale, the coupling terms are small and two independent parabolic equations (1) describe propagation of each component of the field. There is no energy exchange between components, and if the initial condition consists of a linear polarized wave, it will remain linearly polarized.

The above-mentioned papers followed the traditional approach and used uncoupled propagation equations. The second moment of the vector field was described by the

2 \times 2

coherence matrix

Γ_{i j} (R, ρ) = 〈 u_{i} (R + ρ / 2) u_{j}^{*} (R - ρ / 2) 〉

. The parabolic equations for the coherence matrix components

Γ_{i j} (R, ρ)

are exactly the same as Eq. (28) for a scalar wave, and the solution is given by Eq. (30) for each component

Γ_{i j} (r_{1}, r_{2})

. The only difference with the scalar (single wave) is in initial conditions. Convolution equation (49) is valid for the mutual irradiances

I_{i j} (R) = Γ_{i j} (R, 0)

. Again, similar to the instantaneous field components, each component of the coherence matrix propagates independently of the others.

The connection between the field components in the above-mentioned papers was introduced through the initial coherence matrix only, but not by the coupling of the propagation equations. Therefore, all the effects that are claimed to be related to the wave polarization are associated with the mutual coherence of two scalar waves and its evolution in the propagation process. Results are likewise applicable for propagation of the partially coherent scalar beam pairs of arbitrary nature. In practice, for such beam pairs (or multiple beam arrays), individual beams can be discriminated by some measurable parameters such as small frequency shift, time delay, or geometry, but not necessary polarization.

5. SCINTILLATION

A. Weak and Strong Turbulence Conditions

It is very common in the literature to use phrases like “weak turbulence,” “strong fluctuation,” “strong scattering,” etc. to describe propagation conditions. The so-called Rytov variance

σ_{I}^{2} = 1.23 C_{n}^{2} k^{7 / 6} k^{11 / 6}

is very often used to define weak or strong scintillation conditions. We discuss two statistics of the propagating beam waves where it is possible to define the regimes of weak and strong turbulence impact on the propagating wave. The

(N, q)

parameters plane is used to show what the actual conditions of the weak and strong turbulence are. Parameter

N = k a^{2} / L

is the Fresnel number for the aperture or initial beam width, and was introduced in Eq. (22). Parameter

q = k r_{C}^{2} / L

is the Fresnel number for the wave coherence radius (25). Note that parameters

N

and

σ_{I}^{2}

can be used alternatively since

q \propto σ_{I}^{- 12 / 5}

. However, the boundaries between the weak and strong turbulence effects have a simpler form on the

(N, q)

plane than on the

(N, σ_{I}^{2})

plane.

Figure 6 shows domains of the weak and strong beam spread for different focusing properties of the beam [5

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

]. Clearly, a single parameter cannot describe the variety of domain shapes that are present here. Figure 7 shows domains of the weak and strong axial scintillation for beams with different focusing conditions. Charts are based on the complete asymptotic analysis of the beam SI performed in [5

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

]. It is only for the collimated beam that the Rytov variance determines the scintillation conditions. It is worth mentioning that in many cases, weak/strong conditions for the beam spread and scintillation are different. Figure 8(a) shows the domains of the weak and strong scintillation near the focal plane of the beam on the ground to space propagation path. In this case scintillation can be both weak and strong for

σ_{I}^{2} < 1

as well as for

σ_{I}^{2} > 1

[19

M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187 (2010). [CrossRef]

]. Figure 8(b) shows weak and strong power flux fluctuation domains for the aperture-averaged scintillations [9

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991). [CrossRef]

]. It is not surprising that aperture-averaged scintillation can be weak for

σ_{I}^{2} > 1

. More interesting is that in some cases weak flux fluctuations are described by the strong scintillation theory [9

M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991). [CrossRef]

Fig. 6. Domains of the weak and strong beam spread at the

(N, q)

plane. Focusing parameter

d = 1 - L / F

Fig. 7. Domains of the weak and strong beam scintillation at the

(N, q)

plane for various focusing conditions. Focusing parameter

d = 1 - L / F

Fig. 8. Domains of the weak and strong beam scintillation at the

(N, q)

plane. (a) Vicinity of focus for ground to space path. Parameter

θ ≪ 1

is the ratio of the path length

L

to the effective thickness of the turbulent atmosphere. (b) Aperture-averaged scintillation from the point source.

B. Focused Beams Scintillation Paradox

Calculations of the beam SI for the focal point of a Gaussian beam with initial size

a

in the first order of the perturbation theory (or Rytov’s smooth perturbation) give the following well-known result:

σ_{I (1)}^{2} = 2 π \int_{0}^{L} d z z^{2} {(1 - \frac{z}{L})}^{2} \iint κ^{2} d^{2} κ Φ_{n} (κ, z) \exp (- κ^{2} a^{2} {(1 - \frac{z}{L})}^{2}) .

(56)

For the Kolmogorov spectrum (18), Eq. (56) simplifies to

σ_{I (1)}^{2} = 0.60 a^{- 7 / 3} \int_{0}^{L} d z C_{n}^{2} (z) z^{2} {(1 - \frac{z}{L})}^{2} = O (q^{- 5 / 6} N^{- 7 / 6}) .

(57)

Equation (57) predicts a fast decrease of scintillation with increasing beam size, when the beam becomes focused more sharply. This paradoxical result was obtained at least 40 years ago and is still controversial in the modern literature. While wide, sharply focused beams are not seen in many applications; reciprocity indicates that the imaging PSF will have the same SI, and this is more concerning for applications [20

M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W (2011). [CrossRef]

Complete explanation of this paradox was given in [5

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

], where it was shown that in the vicinity of the beam focus the second-order perturbation term

σ_{I (2)}^{2} = 0.33 k^{4} a^{10 / 3} {[\int_{0}^{L} d z C_{n}^{2} (z) {(1 - \frac{z}{L})}^{5 / 3}]}^{2} = O (q^{- 5 / 3} N^{5 / 3})

(58)

predicts the fast growth of the SI with the beam size and successfully competes with the first-order term. The second-order term overcomes the first-order one when

N > q^{17 / 5}

in the so-called

D_{1}

domain. A recent scintillation study [21

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011). [CrossRef]

] based on the direct simulation of the beam wave propagation confirmed the presence of the

D_{1}

domain. A similar situation occurs in the strong scintillation regime where the correction terms are related to scattering from the coherence channels, and second-order scattering dominates for larger beam sizes.

6. SHORT-EXPOSURE IMAGING

Models of imaging through turbulence distinguish between the long-exposure (LE) and short-exposure (SE) PSF. The LE PSF is well understood, and is just a product of the optical system PSF by the average PSF given by Eq. (51). The SE PSF is usually defined as the average of the instantaneous PSF after random displacement of the image points (image warping) is removed. It is expected that the image warp removal improves the resolution of the turbulence imaging, and possibly creates super resolution opportunities [22

M. I. Charnotskii, “Superresolution in dewarped anisoplanatic images,” Appl. Opt. 47, 5110–5116 (2008). [CrossRef]

Fried was the first to propose a model for the SE PSF [23

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]

]. Fried’s SE PSF is based on the phase screen propagation model and assumes the statistical independence of the phase tilt from the higher-order aberrations. It has a form that is very similar to the LE PSF (50):

{PSF}_{SE}^{Fr} (r) = \frac{1}{4 π^{2}} \iint d^{2} ρ \exp [- \frac{1}{2} D_{SE}^{Fr} (ρ) + i \frac{k}{L} ρ \cdot r], D_{SE}^{Fr} (ρ) = 2 r_{C}^{- 5 / 3} [ρ^{5 / 3} - V ρ^{2}],

(59)

but differs by Fried’s SE structure function

D_{SE}^{Fr}

. According to [23

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]

], coefficient

V > 0

depends on the Fresnel number

N

. Recently an attempt was made in [24

D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011). [CrossRef]

] to improve Fried’s SE PSF by introducing the dependence of the coefficient

V

on the turbulence strength.

Figure 9 shows Fried’s structure function for several values of the Fresnel number

N

as dashed lines. It is obvious that Fried’s model violates the principal property of the structure function: it becomes negative for larger separations. The negative values of

D_{SE}^{Fr}

are an unavoidable consequence of the functional form of Eq. (59). For large separations, the

ρ^{2}

term in

D_{SE}^{Fr}

overcomes the

ρ^{5 / 3}

term, and makes the structure function negative. While the negative values of

D_{SE}^{Fr}

do not happen inside the imaging aperture, the invalid trend toward the negative values starts at the half point of the working separation range, and affects the large part of the aperture area.

Fig. 9. Dependence of normalized structure functions on the separation and Fresnel number

N

. Solid lines, SE structure functions of [25

M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993). [CrossRef]

]. Dashed lines, Fried’s [23

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]

] model [Eq. (57)]. Dotted line, LT structure function. Crosses,

N = 0.1

; squares,

N = 0.3

; diamonds,

N = 1

; and unmarked line,

N > 10

Another property that should be expected from any SE structure function is that in the case of pure tilt perturbation we have

D_{SE} (ρ) \equiv 0

, thus providing a diffraction-limited image [20

M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W (2011). [CrossRef]

]. Fried’s model exhibits this property. Unfortunately, the SE model of [24

D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011). [CrossRef]

] cannot guarantee it, since it is based on extrapolation of the limited numerical data.

Almost 15 years ago the ST PSF model was proposed in [25

M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993). [CrossRef]

], which uses several versions of the SE structure functions of various complexities. This model is based on the rigorous solution of the propagation problem, and meets all the theoretical expectations regarding the nonnegative values of the structure function, and suppression of the low-frequency part of the turbulent spectrum. While initial derivation of this model, based on the Feynman path-integral technique [2

], is fairly complex, the final equations are practicable. Some examples of the SE structure function calculations are shown as solid lines in Fig. 9. They reveal some interesting nonmonotonous dependence on the separation parameter that can be used to optimize the SE imaging by adjusting the aperture size.

7. CONCLUSION

We reviewed some important results of the theory of wave propagation in turbulent atmosphere. None of these results are original to this paper. Some have been originally published in Russian 20–30 years ago. Most of these important findings are not covered in the currently available texts. We hope that the points we discussed will be helpful for engineers and scientists working in the field of the wave propagation through turbulence.

REFERENCES

1.	S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).
2.	M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics , Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.
3.	V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).
4.	V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]
5.	M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]
6.	V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).
7.	V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).
8.	M. I. Charnotskii, “Turbulence effects on the imaging of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996). [CrossRef]
9.	M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991). [CrossRef]
10.	M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).
11.	L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
12.	M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010). [CrossRef]
13.	V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
14.	S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980). [CrossRef]
15.	G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006). [CrossRef]
16.	M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203 (2009).
17.	M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).
18.	M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010). [CrossRef]
19.	M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187 (2010). [CrossRef]
20.	M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W (2011). [CrossRef]
21.	M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011). [CrossRef]
22.	M. I. Charnotskii, “Superresolution in dewarped anisoplanatic images,” Appl. Opt. 47, 5110–5116 (2008). [CrossRef]
23.	D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]
24.	D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011). [CrossRef]
25.	M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(030.1670) Coherence and statistical optics : Coherent optical effects
(030.7060) Coherence and statistical optics : Turbulence
(110.4850) Imaging systems : Optical transfer functions
(110.0115) Imaging systems : Imaging through turbulent media

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: November 10, 2011
Revised Manuscript: January 18, 2012
Manuscript Accepted: January 31, 2012
Published: April 18, 2012

Virtual Issues
Vol. 7, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Mikhail Charnotskii, "Common omissions and misconceptions of wave propagation in turbulence: discussion," J. Opt. Soc. Am. A 29, 711-721 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=josaa-29-5-711

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References

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).
M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E. Wolf, eds. (North-Holland, 1993), pp. 205–268.
V. I. Gelfgat, “Reflection in a scattering medium,” Sov. Phys. Acoust. 22, 65–66 (1976).
V. P. Lukin and M. I. Charnotskii, “The reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982). [CrossRef]
M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]
V. P. Lukin and M. I. Charnotskii, “Reverse wave propagation in a randomly-inhomogeneous medium,” Russ. Phys. J. 28, 894–904 (1985).
V. U. Zavorotnyi, “Origin of intensity fluctuations in the image of an incoherent object observed through a turbulent medium,” Opt. Spectrosc. 65, 575–576 (1988).
M. I. Charnotskii, “Turbulence effects on the imaging of an object with a sharp edge: asymptotic technique and aperture-plane statistics,” J. Opt. Soc. Am. A 13, 1094–1105 (1996). [CrossRef]
M. I. Charnotskii, “Asymptotic analysis of flux fluctuation averaging and finite-size source scintillations in random media,” Waves Random Media 1, 223–243 (1991). [CrossRef]
M. I. Charnotskii, “Coupling turbulence-distorted wave front to fiber: Wave propagation theory perspective,” Proc. SPIE 7814, 78140I1 (2010).
L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Media 11, 271–291 (2001).
M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q (2010). [CrossRef]
V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980). [CrossRef]
G. J. Baker, “Gaussian beam weak scintillation: low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006). [CrossRef]
M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203(2009).
M. I. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7865, 786502 (2010).
M. I. Charnotskii, “Coherence of beam arrays propagating in the turbulent atmosphere,” Proc. SPIE 7685, 76850Q(2010). [CrossRef]
M. I. Charnotskii, “Beam scintillation for the ground-to-space propagation,” J. Opt. Soc. Am. A 27, 2169–2187(2010). [CrossRef]
M. I. Charnotskii, “Statistics of the point spread function for imaging through turbulence,” Proc. SPIE 8014, 80140W(2011). [CrossRef]
M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011). [CrossRef]
M. I. Charnotskii, “Superresolution in dewarped anisoplanatic images,” Appl. Opt. 47, 5110–5116 (2008). [CrossRef]
D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). [CrossRef]
D. H. Tofsted, “Reanalysis of turbulence effects on short-exposure passive imaging,” Opt. Eng. 50, 016001 (2011). [CrossRef]
M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993). [CrossRef]

Al-Habash, M. A.
Andrews, L. C.
Baker, G. J.
Charnotskii, M. I.
Fried, D. L.
Gelfgat, V. I.
Gozani, J.
Hopen, C. Y.
Kravtsov, Yu. A.
Lukin, V. P.
Phillips, R. L.
Rytov, S. M.
Tatarskii, V. I.
Tofsted, D. H.
Wandzura, S. M.
Zavorotny, V. U.
Zavorotnyi, V. U.

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